0
$\begingroup$

Let $V=\mathbb{Z}/(3) \oplus \mathbb{Z}/(3) \oplus \mathbb{Z}/(9)$.

  1. How many submodules with $3$ elements does $V$ have?

Because $\phi(3)=2$, each subgroup of order $3$ has $2$ elements of order $3$. Since no $2$ cyclic subgroup can have an element of order $3$ in common, there $6\div 2=3$ cyclic subgroups. Note that there are no non-cyclic subgroups with order $3$ because $\mathbb{Z}_3$ is always cyclic.

  1. How many of the submodules $W$ of $V$ with $3$ elements have a complementary direct summand, i.e., are such that there exists a submodule $W'$ of $V$ with $V=W\oplus W'$?

I want to say that $W'=\mathbb{Z}_3\oplus \mathbb{Z}_9$ but it feels like I am guessing. Also how can I determine how many submodules there are with this property?

  1. How many cyclic submodules with $9$ elements does $V$ have?

Because $\phi(9)=6$, each subgroup of order $9$ has $6$ elements of order $9$. Since no $2$ cyclic subgroups can have an element of order $9$ in common, there are $54\div 6=9$ cyclic subgroups.

  1. How many noncyclic submodules with $9$ elements does $V$ have?

The noncyclic subgroups with order $9$ are isomorphic to $\mathbb{Z}_3\oplus \mathbb{Z}_3$. i.e. In a non-cyclic subgroups of order $3$ each non-identity element is of order $3$.

Where do I go from here?

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.